# Proof about operations on sets

• chocolatelover
In summary, the conversation discusses proving that the union of two sets, each indexed by all real numbers, is a subset of the union of the two sets indexed by real numbers. A hint is given to prove this by showing that any element in the left hand side must also be in the right hand side. The conditions on the left involve the sets Ar and Br, which must be unions of each other and subsets of the other set.
chocolatelover

## Homework Statement

Let the set Ar: r is an element of all real numbers and the set Br: r is an element of all real numbers be two indexed families of sets.

Prove that (upside U r is an element of the reals Ar) U (upside U r is an element of the reals Br) is a subset of upside U r is an element of the reals (ArUBr).

## The Attempt at a Solution

Could someone please give me a hint? I know that Ar and Br have to be unions of each other and at the same they are subsets of the other one, but I don't really know how I would go about proving this.

Thank you much

That's kind of hard to read. But if x is an element of the left hand side, then it's either in all of the A_r or it's in all of the B_r. If x is in the right hand side then it's in either A_r or B_r for all r. Assume each of the conditions on the left and prove they imply the right.

Thank you very much

Regards

## 1. What are the basic operations on sets?

The basic operations on sets are union, intersection, difference, and complement.

## 2. How is the union of two sets represented?

The union of two sets A and B is represented as A ∪ B.

## 3. What is the result of the intersection of two sets?

The intersection of two sets A and B is the set of elements that are common to both A and B, and is represented as A ∩ B.

## 4. How is the difference between two sets defined?

The difference between two sets A and B is the set of elements in A that are not in B, and is represented as A - B.

## 5. What is the complement of a set?

The complement of a set A is the set of all elements in the universal set that are not in A, and is represented as A'.

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